L(s) = 1 | + (−0.743 + 0.669i)2-s + (−0.994 − 0.104i)3-s + (0.104 − 0.994i)4-s + (0.809 − 0.587i)6-s + (0.587 + 0.809i)8-s + (0.978 + 0.207i)9-s + (−0.978 + 0.207i)11-s + (−0.207 + 0.978i)12-s + (−0.951 + 0.309i)13-s + (−0.978 − 0.207i)16-s + (0.406 − 0.913i)17-s + (−0.866 + 0.5i)18-s + (−0.104 − 0.994i)19-s + (0.587 − 0.809i)22-s + (0.743 − 0.669i)23-s + (−0.5 − 0.866i)24-s + ⋯ |
L(s) = 1 | + (−0.743 + 0.669i)2-s + (−0.994 − 0.104i)3-s + (0.104 − 0.994i)4-s + (0.809 − 0.587i)6-s + (0.587 + 0.809i)8-s + (0.978 + 0.207i)9-s + (−0.978 + 0.207i)11-s + (−0.207 + 0.978i)12-s + (−0.951 + 0.309i)13-s + (−0.978 − 0.207i)16-s + (0.406 − 0.913i)17-s + (−0.866 + 0.5i)18-s + (−0.104 − 0.994i)19-s + (0.587 − 0.809i)22-s + (0.743 − 0.669i)23-s + (−0.5 − 0.866i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3067296237 - 0.1899497710i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3067296237 - 0.1899497710i\) |
\(L(1)\) |
\(\approx\) |
\(0.4755559150 + 0.01501079515i\) |
\(L(1)\) |
\(\approx\) |
\(0.4755559150 + 0.01501079515i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.743 + 0.669i)T \) |
| 3 | \( 1 + (-0.994 - 0.104i)T \) |
| 11 | \( 1 + (-0.978 + 0.207i)T \) |
| 13 | \( 1 + (-0.951 + 0.309i)T \) |
| 17 | \( 1 + (0.406 - 0.913i)T \) |
| 19 | \( 1 + (-0.104 - 0.994i)T \) |
| 23 | \( 1 + (0.743 - 0.669i)T \) |
| 29 | \( 1 + (0.809 + 0.587i)T \) |
| 31 | \( 1 + (-0.913 - 0.406i)T \) |
| 37 | \( 1 + (0.207 - 0.978i)T \) |
| 41 | \( 1 + (-0.309 - 0.951i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.406 - 0.913i)T \) |
| 53 | \( 1 + (0.994 + 0.104i)T \) |
| 59 | \( 1 + (0.669 - 0.743i)T \) |
| 61 | \( 1 + (-0.669 - 0.743i)T \) |
| 67 | \( 1 + (-0.406 + 0.913i)T \) |
| 71 | \( 1 + (-0.809 - 0.587i)T \) |
| 73 | \( 1 + (-0.207 - 0.978i)T \) |
| 79 | \( 1 + (-0.913 + 0.406i)T \) |
| 83 | \( 1 + (-0.587 - 0.809i)T \) |
| 89 | \( 1 + (0.669 + 0.743i)T \) |
| 97 | \( 1 + (-0.587 + 0.809i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−27.535216293742584759101068430207, −27.06982386648184442925526519733, −25.961672817431794373576863069896, −24.82221763370181810025530325018, −23.61825710324789273477475393870, −22.673673995248028572815688941563, −21.55651025068121055856131931605, −21.07264317804708520678177433170, −19.6836236447599594465671453777, −18.743169300768324155388318321259, −17.87120664214459287585569180802, −17.00153385128223348417088363652, −16.23222601702683337659379896868, −15.037072468642195184312553241341, −13.15889125056483156166394130117, −12.42548181902583971042252884728, −11.436985058233374260216744125914, −10.39513634907500374597058581164, −9.82414503132680088743886583081, −8.21976416848562029927775066254, −7.25803893873458065180534125756, −5.8122589763346258078826476797, −4.51730786940635794139665925600, −3.03375131263195764569041214237, −1.39840650279236173281638855105,
0.44312884633739637845855155136, 2.2793338255288206203079614697, 4.79302943591295084534240857387, 5.431445663510647842423268257674, 6.89686312323490525413222250016, 7.43885651830127458819160951535, 8.99156513593260868599286449539, 10.11775641079642205488349537450, 10.92646600573950232027884506584, 12.09877824210947254547826075568, 13.349313098946007476558138376548, 14.71949210050279437838873461001, 15.77922412118321518605448333611, 16.553343484370995173405559310830, 17.476681834357311352345222810249, 18.2780129138789156265500616627, 19.088370103766340243340205419338, 20.34023267295774953096387642347, 21.6349741810522292394735383615, 22.77181874637080435810677789572, 23.603762187996360658859258975733, 24.33136254583339749008737382406, 25.322843095541655419421399441402, 26.498858185785677208968446112921, 27.22534082131678535326231047353